Papers

We consider vector mean-field spin glass models with independent and identically distributed spins without assuming the covariance function is convex. We identify the limit free energy as the minimum of a functional that can be written as the sum of a relative entropy and the solution of a Hamilton–Jacobi equation. The proof of this result relies on the observation that when the spin configurations of the system are constrained to have a given empirical measure, the corresponding free energy approximately satisfies a Hamilton–Jacobi equation with a uniformly bounded error.

We develop a rate-transfer method that converts finite-time Lyapunov estimates for an associated stochastic differential equation (SDE) into finite-time convergence bounds for stochastic gradient algorithms. Under generic moment and smoothness conditions, the convergence analysis of the stochastic gradient method reduces to a calculus argument verifying a Lyapunov generator inequality for the SDE. The method recovers established rates for classical stochastic gradient methods and provides a way to establish convergence rates for new optimization algorithms through calculus arguments applied to their associated SDEs.

We consider the problem of estimating the expectation of a function (observable) with respect to a high-dimensional probability distribution, under the assumption that a low-dimensional collective variable map captures all the directions of slow mixing for the distribution. More precisely, under the assumption that the fibers of the collective variable map satisfy a log-Sobolev inequality, we establish non-asymptotic relative-error guarantees for this problem using the eigenvector method for umbrella sampling (EMUS). Our analysis applies to both log-Lipschitz observables and indicator of convex sets, and gives complexity bounds that are polynomial in regularity parameters and inverse relative accuracy, and exponential in the dimension of the collective variable space. As a relative error guarantee, our bounds apply to rare event estimation. We consider Gaussian biasing windows defined on a low-dimensional collective variable space and analyze a modified sparse matrix version of EMUS. The bounds quantify the effect of the map through mixing of the conditional distribution and fluctuation of the observable along the fibers.

We construct an approximate sampler for finite doubly free-boundary Macdonald rail-yard processes with Macdonald, boundary, and edge parameters in $(0,1)$. The construction allows arbitrary finite sequences of additions and removals of horizontal or vertical strips and all four boundary types. It reverses a finite commutation–reflection history from a partition sequence whose entries are equal, with the common partition weighted by both boundaries. We derive local reconstruction kernels from the skew Cauchy, Littlewood, and branching identities. Bulk and boundary updates combine integer increments with conditional shape sampling at a prescribed size; same-sign swaps use finite conditional resampling. For $K$ reflection periods and an initial partition-size cutoff $M$, the algorithm terminates almost surely and samples the resulting approximating law exactly. We prove total variation convergence to the target process as $K,M\to\infty$ and give explicit bounds for both truncation errors, decaying geometrically in the cutoffs for fixed input parameters. Under fixed parameter bounds, we also bound the expected operation count of direct enumeration in a unit-cost model. We further construct an exact rejection sampler for the initial partition law without a size cutoff or evaluation of its normalizing constant. Using this sampler removes the size-truncation error, leaving only the error from the finite reflection history.

We show that the guaranties of the Gram-Schmidt Walk Algorithm of Bansal, Dadush, Garg, and Lovett can be achieved by an Ising model. Given vectors $v_1, \ldots, v_n$ of norm less than 1, we give a closed form for a centrally symmetric distribution on a vector $\sigma \in \{\pm 1 \}^n$ so that the signed sum $\sum_j \sigma(j) v_j$ is $1$-subgaussian. We show that the Glauber dynamics for this model mix rapidly provided that the norms of the input vectors are bounded below 1 by an inverse-polynomial. The proofs employ the theory of Lorentzian polynomials, and we use these techniques to show fast mixing bounds for other spin systems.

Sampling from high-dimensional, heavy-tailed distributions is a fundamental challenge in computational statistics, as many standard Markov chain Monte Carlo (MCMC) methods mix poorly in such settings. Recently, Stereographic MCMC [Yang et al., 2024] and the Sub-Cauchy Projection Sampler [Grazzi et al., 2026] have been shown to perform successfully on such tasks. However, establishing their non-asymptotic convergence properties remains an important open problem. In this work, we fill this gap by establishing non-asymptotic upper bounds on the mixing time of the stereographic projection and sub-Cauchy projection samplers. Our results demonstrate that, under certain conditions on the target and initial distributions, the mixing time is polynomial in dimension for a broad class of distributions, including light- and heavy-tailed cases. Motivated by the theoretical analysis, we further establish a new weighted isoperimetric inequality that extends the classical version for log-concave distributions to the heavy-tailed setting with optimal dimension dependence. The proof techniques provide new insights into the geometric properties of heavy-tailed distributions that govern rapid mixing in high dimensions.

In the past two decades, the Bayesian approach to inverse problems in infinite dimensional spaces has seen significant developments in theory, Markov chain Monte Carlo (MCMC) methods, and in applications. The Bayesian approach in finite dimension produces posterior distributions of finite dimensional random variables. In infinite dimensions, the desired objects are random fields and their laws as measures on Hilbert or Banach spaces. There are many textbooks and tutorials available that explain Bayesian methods and MCMC in finite dimensions. To enter the field of Bayesian inverse problems in infinite dimensions, a newcomer has to engage with a quite diverse and extensive literature. The purpose of this article is to provide a basic path to the subject that is easily accessible to a reader with general knowledge in probability and mathematical analysis. We start with a swift introduction to the finite dimensional case. The infinite dimensional case is introduced for Gaussian priors on Hilbert spaces and Gaussian-type likelihood functions. In this situation, mere continuity of the forward map suffices to prove that the posterior measure is well-defined. We also give some new and simple stability results. Two application examples are elaborated, one finite dimensional, one infinite dimensional.

In this paper, we study a new variant of the ants walk: a model of reinforced random walks for the phenomenon of a colony of ants being able to find the shortest path between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes in a finite graph. The ants perform successive random walks on this graph, started at the nest and stopped when they first hit the food source; the transition probabilities of each random walk depends on the trajectories of the previous ants, through some edge-weights that represent the amount pheromones deposited on each edge. In our new variant of the model, the ant (1) forgets loops made on the way to the food as soon as they are made, and (2) comes back to the nest following this loop-erased version of its forward trajectory, and deposits pheromone on the edges it crosses by doing so (i.e. adds one to these edges' weights). We say that the ants "find the shortest path" if, almost surely as the number of ants tends to infinity, the proportion of ants that travel along a graph-geodesic path from the nest to the food tends to one and the proportion of ants that travel along the remaining edges tends to zero. We are able to prove that this new variant of the model finds the shortest path when the graph distance between the nest and the source is at most 2. The strength of our result is that there is no other assumption on the shape of the graph. We prove this result using stochastic approximation theory, the electrical network method for random walks on weighted graphs, and the theory of weighted uniform spanning trees (WUSTs). In particular, the link with WUSTs is unique to the model studied in this paper due to the way the ant erases loops on the outward journey.

Automating AI research creates a positive feedback loop between software capability and research effort. We study a stochastic realisation of this loop in which independent, positive, multiplicative improvements arrive at a rate proportional to a power of current capability. Within this model, randomness leaves the deterministic explosion threshold unchanged but replaces the blow-up date by an explicit distribution. For equal-sized improvements the explosion time is a geometrically weighted sum of independent exponentials; its $q$-series representation gives its transforms, moments and tail behaviour, including a two-term small-time asymptotic. For Lomax-distributed improvement sizes the explosion time is exactly Gamma distributed. Self-similarity implies that, with parameters known, the relative uncertainty about the remaining time does not vanish as explosion approaches. We formulate a fixed-sample test between constant and geometrically increasing idea-arrival rates. If the baseline rate and the alternative log-rate increment $\ell$ are known, the minimum sample size at error level $\alpha$ is asymptotic to $(12z_\alpha^2)^{1/3}\ell^{-2/3}$, where $z_\alpha$ is the upper-$\alpha$ standard normal quantile. Uncertainty about the growth exponent near its critical value produces a $1/t$ tail with an explicit constant, and hence an infinite conditional mean waiting time, although explosion is certain given a positive exponent. Separate extensions show that a fixed deployment delay prevents explosion in the specified delayed-feedback models, that power-law diversion of effort shifts the growth threshold, and that in a deterministic oversight race a fixed allocation share does not change the asymptotic regime, except at a critical equality of exponents. Numerical illustrations, which are not empirical forecasts, show how timing and detectability depend on the granularity of improvements.

The Sachdev–Ye–Kitaev (SYK) model is a quantum many-body model of Majorana fermions with all-to-all random interactions. It has attracted wide interest in condensed matter and high-energy physics as a solvable model of non-Fermi liquids and holography. Fix an even integer $q\geq4$, and let $Z_N(\beta)$ be the partition function of the Gaussian $q$-body SYK model. We prove that for all sufficiently small inverse temperatures $\beta > 0$, there exists $\rho=\rho(\beta)>0$ such that \[ N^{q/2 -1}(\log Z_N(\beta)-\mathbb{E}[\log Z_N(\beta)]) \Longrightarrow \mathcal{N}(0,\rho^2). \] In addition to the mathematical results, we discuss how the proofs were developed and refined through interaction between human researchers and AI, and we describe the subsequent formalization of the main arguments in Lean 4. This work thus serves as a case study of a research workflow that integrates human insight, AI-assisted exploration, iterative proof refinement, and formal verification.

We prove that, for every $c>0$, the image of the generalized Markov-Krein transform on probability measures with finite logarithmic moment is closed under classical additive convolution and under the Mellin convolution. This resolves the positivity conjecture for the additive high-temperature convolution, a one-parameter interpolation between classical and free convolution arising in random matrix theory and representation theory in the regime $N\to\infty, N\beta_N/2\to c$. Our proof relies on a convolution theorem for universally prestarlike functions.

Time-inconsistent stochastic control admits a natural game-theoretic interpretation in which successive decision dates are represented by different selves. In the Markovian framework of Björk-Khapko-Murgoci, the corresponding continuous-time equilibrium HJB system can be motivated by discrete subgame-perfect games, but the general discrete-to-continuous passage is not automatic. We establish a quantitative convergence theorem for a smooth uniformly elliptic class in which the control may enter both drift and diffusion. Using the exact sample-and-hold transition law on each time step, we construct, near a verified continuous equilibrium branch, a unique regular discrete Markov subgame-perfect equilibrium and prove first-order convergence of the continuation objects in an anisotropic $C^{2+\beta}$ topology and of the equilibrium policies in $C^0$. The key stability mechanism is structural: the discrete first-order condition forces the differential feedback correction in the frozen linearization to be nilpotent, which yields mesh-uniform block estimates through a two-norm dyadic Schauder argument despite the second-order coupling caused by controlled volatility. We also formulate a conditional transfer principle for structured polynomial-growth classes, with error-space stability stated as a separate hypothesis.

It is shown that the random prime Dirichlet series $ X(s)=\sum_{p\in \mathcal{P}}\frac{\xi_p}{p^{s+1/2}} $ can be almost surely approximated by a Brownian motion, where $ \{\xi_p\}_{p\ge 1} $ are independent and identically distributed random variables with zero mean and finite variance, and $ \mathcal{P} $ is the set of prime numbers. As a consequence, the functional law of the iterated logarithm, limit laws of the iterated logarithm, and Chung's law of the iterated logarithm for $ X(s) $ are obtained. It is also shown that the random Dirichlet series $ Y(s)=\sum_{k=2}^{\infty} \frac{(\log k)^{\alpha}}{k^{s+1/2}} $ with $ \alpha=-1/2 $ admits the same approximation as $ X(s) $, and a direct relationship between $ X(s) $ and $ Y(s) $ with $ \alpha=-1/2 $ is established. The strong invariance principle for $ Y(s) $ with $ \alpha>-1/2 $ is also obtained.

We identify the limit free energy of mean-field multi-species spin glasses with +/-1 spins, allowing for the presence of external fields, and without assuming any convexity property of the covariance function of the Hamiltonian. The limit is given by an "uninverted" variational formula over Brownian martingales. The proof consists in first identifying the limit free energy of configurations with a fixed magnetization. Once the magnetization is fixed, we can show the convergence of the enriched free energy to the solution of a Hamilton-Jacobi equation. The magnetization constraint is also used to show the convexity of the initial condition in this equation, and this yields a variational representation of the solution, as a Hopf formula. Convex duality then turns this formula into an optimization over martingales whose initial values are the prescribed magnetizations.

We consider the oriented site percolation model in a random environment introduced by Kesten, Sidoravicius and Vares (2022), in which the horizontal lines of the lattice are independently declared bad or good, sites being open with a probability that depends only on the type of their line. We study the boundary case where the bad lines are critical and the good ones supercritical, so that the model is critical oriented site percolation enhanced along a random set of good lines. We prove that percolation occurs for any positive density of the good lines. An arbitrarily sparse set of arbitrarily weakly enhanced lines suffices, the density of good lines and the strength of the enhancement being decoupled. The proof adapts the renormalization scheme of Archer, Hartarsky, Kolesnik, Olesker-Taylor, Schapira and Valesin (2026). We show that the edge of the process travels at a positive speed, which yields crossing estimates for tilted boxes; during runs of bad lines, we rely on the box-crossing property of Duminil-Copin, Tassion and Teixeira (2018); and the renormalized model dominates the oriented percolation with geometric defects of Hilário, Sá, Sanchis and Teixeira (2026).

This work is a step towards a general understanding of fluctuations for systems of particles described at large scale by fractional diffusion. First, we derive formally a Large Deviation Principle for the empirical measure of mean-field interacting particles with a Lévy-type noise. In the case of independent particles, this unveils a generalized gradient flow structure at large scales. Second, we derive and solve numerically a stochastic Partial Differential Equation describing the fluctuating hydrodynamics of this system. The results compare well with direct numerical integration of the particles' dynamics.

We investigate the effect of initially enlarging a Brownian filtration by its terminal value on backward stochastic differential equations (BSDEs) and reflected BSDEs (RBSDEs). We establish the Brownian innovation representation and the predictable representation property in the enlarged filtration, highlighting the singular information drift arising from the semimartingale decomposition of the original Brownian motion. We derive transfer identities under suitable integrability conditions and provide explicit counterexamples showing that the resulting singular equations may fail to admit unique square-integrable solutions, including in the reflected case. We further develop a conditional parametrization of regular BSDEs and establish a conditional comparison principle for Snell envelopes, illustrating both the strict value of additional information and the failure of general pathwise domination. An exact projection identity is also obtained for zero-driver terminal payoffs. Finally, we discuss possible extensions to irregular obstacles and more general information structures.

We give an estimate for the distribution tail of the passage time between $0$ and $\infty$ in first-passage percolation on $\mathbb Z^d$, when the weight distribution is bounded and has a supercritical atom at $0$. A special case involves the number of disjoint cutsets between $0$ and $\infty$ in supercritical bond percolation. Besides, we show that the rate function for the upper-tail deviations of the point-to-point passage time is asymptotically polynomial near $0$.

In this article, we study optimal control problems driven by mean-field forward-backward stochastic differential equations (FBSDEs). The aim is two-fold: (i) to provide the unique optimal control; (ii) to establish classical solutions of the associated generalized Hamilton-Jacobi-Bellman (HJB) equations on the space of probability measures. For (i), we study the coupled system arising from the maximum principle, which involves both the state FBSDEs and the adjoint FBSDEs. This system is non-standard and more complicated than that in standard mean-field type control (MFTC) problems. By extending the method of continuation in the coefficients to our framework, we establish the well-posedness result, and subsequently obtain the differentiability of the value functional with respect to the distribution argument — thereby extending existing results for MFTC problems. For (ii), since the flow property does not generally hold for control problems of FBSDEs, standard dynamic-programming arguments cannot be applied directly. We therefore develop probabilistic approaches to the classical solvability of generalized HJB equations under three distinct sets of conditions. The auxiliary processes arising in these arguments differ significantly from those in standard MFTC problems. We establish new connections between the value function and the adjoint processes, derive equations for the relevant decoupling field, and obtain first- and second-order generalized HJB equations, including a coupled system on the space of probability measures. The standard HJB equations for MFTC problems are recovered as special cases. Our approach yields classical solutions without requiring non-degeneracy of the diffusion.

We study the eigenvalues of the negative Dirichlet Laplacian $-\Delta$ in domain $D_{\epsilon,N}$ obtained by removing $N$ Euclidean balls of radius $\epsilon$ from a bounded open set $D\subseteq\mathbb R^d$ ($d\ge2$). It is known that, for the balls centered at points drawn independently from a distribution with probability density $\rho$ and for $N\to\infty$ and $\epsilon\downarrow0$ so that the aggregate Newtonian capacity of the removed balls tends to some $\kappa>0$, the eigenvalues of $-\Delta$ in $D_{\epsilon,N}$ homogenize to those of the Schrödinger operator $-\Delta+\kappa\rho$ in $D$. Here we prove that, for the eigenvalues that are simple in the limit, a multivariate Central Limit Theorem holds in all $d\ge2$ under centering by expected values. Unlike earlier work in $d=3$ by Figari, Orlandi and Teta and, independently, Ozawa from 1980s that was based on resolvent expansions, we rely on martingale representations and one-ball perturbation asymptotic controlled by methods of harmonic analysis.

We use Loewner-Kufarev evolution to decompose the Brownian loop soup into a Poisson point process on Brownian bubbles which we call the Brownian bubble tea driven by $\rho$, where $\rho$ is the driving measure for the Loewner chain. The only assumption on $\rho$ is that the evolving boundaries are locally connected. This generalizes a similar disintegration result of Lawler and Werner to essentially optimal regularity, and includes foliations with finite Loewner-Kufarev energy as well as radial and whole-plane $\mathrm{SLE}_\kappa$ for all $\kappa \geqslant 0$. As applications, we give Brownian loop interpretations of the conformal distortion formula for Loewner-Kufarev energy and the relationships between the Loewner energy, Loewner-Kufarev energy, and the Schwarzian action.

Sampling Gaussian measures restricted to nonlinear manifolds in high dimensions requires a proposal kernel that respects both geometric constraints and dimensional scaling requirements. We introduce tangent preconditioned Crank-Nicolson (pCN), which combines a manifold projection with a cancellation of the ambient Gaussian contribution, leaving only a geometric residual. Considering a shifted sphere with $\sqrt d$ scaling of radius and centre as a tractable benchmark embedded manifold, we prove that uncorrected tangent pCN results in a proposal noise scale of order $d^{-1/6}$ (compared to $d^{-1/2}$ for a tangent random-walk). We establish asymptotic optimal tuning for the geodesic expected squared jump distance which is maximised at an acceptance probability of 0.574 for the family of drift-corrected tangent proposals containing tangent pCN and tangent MALA. We also prove a joint diffusion limit for the parallel and orthogonal coordinates to the sphere's centre vector of tangent pCN. The orthogonal diffusion speed matches the limiting geodesic expected squared jump distance and is maximised at acceptance probability 0.574, whereas the parallel speed is maximised at 0.712, demonstrating that optimal tuning depends on the observable. We then identify an intrinsic precision correction characterised by potential Hessian and Ricci curvature that yields a proposal kernel with dimension-independent fixed step size limit. We discuss a generalisation to arbitrary hypersurfaces and present nondegenerate scaling of tangent pCN applied to conditioned diffusions under resolution refinement.

We establish quantitative hypocoercivity for singular stochastic damped wave equations arising in stochastic quantization. Building on the finite-dimensional argument of Cao, Lu, and Wang [Arch. Rat. Mech. Anal. 2023], we derive a dimension-free criterion that replaces uniform semiconvexity by a log-Sobolev inequality together with exponential lower-tail bounds for the bottom of the Hessian spectrum. We verify this criterion for the two-dimensional $\Phi^4_2$ and sine-Gordon models and for the three-dimensional $\Phi^4_3$ model, through estimates on the lowest eigenvalue of associated singular random Schrödinger operators. Combined with the continuum limit established in a companion paper, this yields exponential $L^2$ convergence to equilibrium for the $\Phi^4_2$ dynamics and for sine-Gordon with $\beta^2<2\pi$. For $\Phi^4_2$, we verify that in a weak-coupling, small-mass regime, optimizing the damping yields a kinetic relaxation rate of at least order $m$. This yields a genuine acceleration over the order-$m^2$ relaxation rate of the overdamped dynamics. The lattice estimates are uniform in the discretization for sine-Gordon throughout $\beta^2<4\pi$, and for $\Phi^4_3$ at fixed volume, including regimes beyond those for which the continuum dynamics is vpresently constructed.

We determine explicit bulk and worst-case total variation cutoff profiles for the configuration law of a two-species Togashi–Kaneko model with a fixed population of $n$ particles and weak mutations. Boundary initial conditions introduce an additional logarithmic growth phase and change both the leading mixing-time coefficient and the limiting cutoff profile. Let $\kappa_1,\kappa_2>0$ be the autocatalytic rate constants and put $\lambda_*=\kappa_1\kappa_2/(\kappa_1+\kappa_2)$. For deterministic initial configurations whose density converges to $m\in(0,1)$, the total variation distance to equilibrium at time $t_n(s)=\frac{\log n}{2\lambda_*}+\frac{s}{\lambda_*}$ converges to $\mathcal G(r(m)e^{-s})$ for every fixed $s\in\mathbb R$, where $\mathcal G(r)=d_{\mathrm{TV}}(\mathcal N(r,1),\mathcal N(0,1))$, $r(m)>0$ is explicit, and $\mathcal N(a,b^2)$ denotes the Gaussian distribution with mean $a$ and variance $b^2$. For the worst-case distance $\mathscr D_n(t)$ over all initial configurations, we prove $\mathscr D_n(T_n+s/\lambda_*)\to\mathcal P(s)$, where $T_n=\left(\frac{1}{\min\{\kappa_1,\kappa_2\}}+\frac{1}{2\lambda_*}\right)\log n$.The profile $\mathcal P$ is the total variation distance between the equilibrium Gaussian and a Gaussian mixture whose random displacement is determined by a gamma-distributed amplitude arising from early growth at the boundary. Unlike the bulk profile, $\mathcal P$ depends on the mutation parameters. Both cutoffs have windows of constant order, and we identify the asymptotically worst boundary configurations.

We investigate whether Rényi divergence can sharpen McDiarmid's inequality and its distribution-dependent refinement induced by KL divergence. Starting from Marton's transportation proof, we show that its transport inequality fails in general when KL divergence is replaced by Rényi divergence of order $0<\alpha<1$. For sums of independent bounded random variables, however, the additivity of Rényi divergence allows us to adapt the argument and derive concentration bounds. For Bernoulli sums, these bounds strictly improve on the distribution-dependent bound induced by KL divergence over a range of deviation thresholds. We also establish a distribution-dependent refinement of Gilardoni's inequality relating total variation to Rényi divergence, providing explicit constants for our general concentration bounds.

We prove that the simulated annealing algorithm efficiently samples from the Sherrington-Kirkpatrick model at inverse temperature $\beta < 1$ to an arbitrary inverse polynomial accuracy in total variation.

In this paper, we study natural determinantal point processes on Stein manifolds defined by canonical sections and plurisubharmonic weight functions. We present a natural condition on the weights under which these processes satisfy a large deviation principle with a good rate function given by the Fenchel–Legendre transform of the pluripotential equilibrium energy. Along the way, we prove some pluripotential theoretic results on Stein manifolds, such as the $C^{1,1}$ regularity of appropriate envelopes. These results are noncompact analogues of results of Berman and Berman–Boucksom on compact Kähler manifolds.

We study a broad class of stochastic transport problems under controlled McKean-Vlasov dynamics, allowing control of both drift and volatility, with weak terminal constraints. In contrast to the classical Schrödinger problem, the reference dynamics depend on the evolving law of the process, leading to nonlinear optimization problems on path space, while the terminal constraint is formulated through weak optimal transport, permitting a flexible treatment of distributional targets. This framework encompasses several problems of current interest, including the mean field Schrödinger problem, martingale optimal transport, and Schrödinger-Bass bridges. We establish three main results. First, we prove the existence of optimal controls under general coercivity, continuity, and convexity assumptions. Second, we derive a convex dual representation that extends weak Kantorovich duality and stochastic optimal transport duality to controlled McKean-Vlasov dynamics. Finally, for the mean field Schrödinger problem, we prove convergence, at the level of value functions, of the associated finite-particle Schrödinger bridge problems to the mean field limit, thereby identifying the nonlinear problem as the exact infinite-particle limit of finite-particle Schrödinger bridges. As a byproduct, we establish the existence of an optimal mean field Schrödinger potential, providing a variational foundation for duality in nonlinear stochastic transport.

We prove that for every $(X,\mathcal{B},\mu,T)$, an aperiodic, probability preserving system and $\mathcal{P}$ a finite measurable partition of $X$, there exists an Alpern Multitower whose base is independent of $\mathcal{P}$. This removes the ergodicity assumption in the original proof of the independent Alpern Lemma of \cite{MR3373823}. As a consequence, we remove ergodicity assumption from recent results on limit theorems in deterministic systems \cite{Kosloff_Volny_2022,kosloffVolnyStableCLT,MR4955410}.

We define multidimensional delta functions as generalized functions by pulling back the Dirac delta function, as well as products of such. We prove various formulas including a local version of the coarea formula and we use delta functions to compute marginal probability distributions on spheres.